Imre Lakatos1922-1974) was a philosopher of mathematics and of science.
Lakatos was born Imre Lipschitz in Hungary. He received a degree in mathematics, physics, and philosophy from the University of Debrecen in 1944. He became an active communist during the second world war.
In 1956, during a time of upheaval in Hungary, Lakatos fled to Vienna, and later reached England. He received a doctorate in philosophy in 1961 from the University of Cambridge. The book Proofs and Refutations, published after his death, is based on this work.
In 1960 he was appointed to a position in the London School of Economics, and remained there until his death. He wrote on the philosophy of mathematics and the philosophy of science. The LSE philosophy of science department at that time included Karl Popper and John Watkins.
Parts of his correspondence with his friend and critic Paul Feyerabend have been published in For and Against Method (ISBN 0226467740).
The book Proofs and Refutations is based on his doctoral thesis. It is largely taken up by a fictional dialogue set in a mathematics class. The students are attempting to prove Euler's formula in algebraic topology, which is a theorem about the properties of polyhedra. The dialogue is meant to represent the actual series of attempted proofs which mathematicians historically offered for the conjecture, only to be repeatedly refuted by counterexamples. Often the students 'quote' famous mathematicians such as Cauchy.
What Lakatos tried to establish was that no theorem of informal mathematics is immune from falsification. This means that we can never know that a theorem is true, only that it has not yet been refuted. (If axioms are given for a branch of mathematics, however, Lakatos claimed that proofs from those axioms were tautological, i.e. logically true.)
Lakatos proposed an account of mathematical knowledge based on the idea of heuristics. In Proofs and Refutations the concept of 'heuristic' was not well developed, although Lakatos gave several basic rules for finding proofs and counterexamples to conjectures. He thought that mathematical 'thought experiments' are a valid way to discover mathematical conjectures and proofs, and sometimes called his philosophy 'quasi-empiricism'.
However, he also conceived of the mathematical community as carrying on a kind of dialectic to decide which mathematical proofs are valid and which are not. Therefore he fundamentally disagreed with the 'formalist' conception of proof which prevailed in Frege's and Russell's logicism, which defines proof simply in terms of formal validity.
On its publication in 1976, Proofs and Refutations became highly influential on new work in the philosophy of mathematics, although few agreed with Lakatos' strong disapproval of formal proof. Before his death he had been planning to return to the philosophy of mathematics and apply his theory of research programmes to it. One of the major problems perceived by critics is that the pattern of mathematical research depicted in Proofs and Refutations does not faithfully represent most of the actual activity of contemporary mathematicians.
Lakatos' contribution to the philosophy of science was an attempt to resolve the perceived conflict between Popper's falsificationism and the revolutionary structure of science described by Kuhn.
Kuhn had described science as consisting of periods of normal science interspersed with periods of great conceptual change, backing up his case with evidence from the history of science. Popper had presented falsificationism as a way to overcome the problem of induction and also to distinguish scientific from non-scientific propositions. Popper's prescription implies a smooth progress from one hypothesis to another as they are falsified and replaced with increasingly bold and powerful hypotheses. This is prima facie at odds with the history of science as described by Kuhn, in which scientists defend their doctrines, even when the evidence against them becomes overwhelming.
The problem for Lakatos was to defend the presumed rationality of scientific method against the apparent impulsiveness of scientists. For Lakatos, science progressed by developing complex research programmes that include testable hypotheses, and also an untestable 'core' of doctrine, which those involved in the research programme would not permit to be falsified.
A research programme (or program) consists of, in Lakatos' terms, a negative heuristic or 'hard core' that is not open to negotiation, and in effect lays down the foundations of the programme. One example given is Newton's three laws of dynamics, which define quantities such as force. These are not open to falsification within the Newtonian system, but are defended at all cost by the positive heuristic, a 'protective belt' of statements that are open to falsification. When falsified, these are replaced by variations that are also falsifiable, but which continue to protect the hard core. Thus a research programme provides a framework within which research can be undertaken with constant reference to presumed first principles which are shared by those involved in the research programme, and without continually defending these first principles.
Lakatos claimed that research programmes could be evaluated by comparing their ability to produce new facts, and by their ability to explain apparent refutations. In effect, a research programme grows as its positive heuristic extends its applicability into new areas. A research programme that is in a state of constantly defending its hard core, and which appears not to be extending itself into new areas, becomes degenerate. Such research programmes are in danger of being superseded by more vigorous competitors.